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Theorem carden2b 8349
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8348 are meant to replace carden 8927 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 8347 . . . . 5  |-  ( (
card `  B )  e.  ( card `  A
)  ->  -.  ( card `  B )  ~~  A )
2 ennum 8329 . . . . . . . 8  |-  ( A 
~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
32biimpa 484 . . . . . . 7  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  e.  dom  card )
4 cardid2 8335 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
53, 4syl 16 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  B )
6 ensym 7565 . . . . . . 7  |-  ( A 
~~  B  ->  B  ~~  A )
76adantr 465 . . . . . 6  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  B  ~~  A )
8 entr 7568 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~~  A )  -> 
( card `  B )  ~~  A )
95, 7, 8syl2anc 661 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  ~~  A )
101, 9nsyl3 119 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  B
)  e.  ( card `  A ) )
11 cardon 8326 . . . . 5  |-  ( card `  A )  e.  On
12 cardon 8326 . . . . 5  |-  ( card `  B )  e.  On
13 ontri1 4912 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( card `  B )  e.  On )  ->  (
( card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
) )
1411, 12, 13mp2an 672 . . . 4  |-  ( (
card `  A )  C_  ( card `  B
)  <->  -.  ( card `  B )  e.  (
card `  A )
)
1510, 14sylibr 212 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  C_  ( card `  B ) )
16 cardne 8347 . . . . 5  |-  ( (
card `  A )  e.  ( card `  B
)  ->  -.  ( card `  A )  ~~  B )
17 cardid2 8335 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
18 id 22 . . . . . 6  |-  ( A 
~~  B  ->  A  ~~  B )
19 entr 7568 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  B )  -> 
( card `  A )  ~~  B )
2017, 18, 19syl2anr 478 . . . . 5  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  ~~  B )
2116, 20nsyl3 119 . . . 4  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  -.  ( card `  A
)  e.  ( card `  B ) )
22 ontri1 4912 . . . . 5  |-  ( ( ( card `  B
)  e.  On  /\  ( card `  A )  e.  On )  ->  (
( card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
) )
2312, 11, 22mp2an 672 . . . 4  |-  ( (
card `  B )  C_  ( card `  A
)  <->  -.  ( card `  A )  e.  (
card `  B )
)
2421, 23sylibr 212 . . 3  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  B
)  C_  ( card `  A ) )
2515, 24eqssd 3521 . 2  |-  ( ( A  ~~  B  /\  A  e.  dom  card )  ->  ( card `  A
)  =  ( card `  B ) )
26 ndmfv 5890 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2726adantl 466 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (/) )
282notbid 294 . . . . 5  |-  ( A 
~~  B  ->  ( -.  A  e.  dom  card  <->  -.  B  e.  dom  card ) )
2928biimpa 484 . . . 4  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  -.  B  e.  dom  card )
30 ndmfv 5890 . . . 4  |-  ( -.  B  e.  dom  card  -> 
( card `  B )  =  (/) )
3129, 30syl 16 . . 3  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  B )  =  (/) )
3227, 31eqtr4d 2511 . 2  |-  ( ( A  ~~  B  /\  -.  A  e.  dom  card )  ->  ( card `  A )  =  (
card `  B )
)
3325, 32pm2.61dan 789 1  |-  ( A 
~~  B  ->  ( card `  A )  =  ( card `  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   (/)c0 3785   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5588    ~~ cen 7514   cardccrd 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7312  df-en 7518  df-card 8321
This theorem is referenced by:  card1  8350  carddom2  8359  cardennn  8365  cardsucinf  8366  pm54.43lem  8381  nnacda  8582  ficardun  8583  ackbij1lem5  8605  ackbij1lem8  8608  ackbij1lem9  8609  ackbij2lem2  8621  carden  8927  r1tskina  9161  cardfz  12049
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